Econometrics – lecture 3 – multiple regression

KTEE 310 FINANCIAL ECONOMETRICS MULTIPLE REGRESSION ANALYSIS: ESTIMATION Chap – S & W Dr TU Thuy Anh Faculty of International Economics MULTIPLE REGRESSION WITH TWO EXPLANATORY VARIABLES: EXAMPLE Q = 1 + 2 L+ 3 K + u The model has three dimensions, one each for Q, L, and K The starting point for investigating the determination of Q is the intercept, 1 1 Q K L MULTIPLE REGRESSION WITH TWO EXPLANATORY VARIABLES: EXAMPLE Q = 1 + 2L+ 3K + u pure L effect 1 1 + 2L Q K L MULTIPLE REGRESSION WITH TWO EXPLANATORY VARIABLES: EXAMPLE Q = 1 + 2 L+ 3 K + u 1 + 3 K pure K effect 1 Q K L MULTIPLE REGRESSION WITH TWO EXPLANATORY VARIABLES: EXAMPLE Q = 1 + 2 L+ 3 K + u 1 + 2L + 3K 1 + 3 K pure K effect Pure L effect 1 combined effect of L and K 1 + 2L Q K L MULTIPLE REGRESSION WITH TWO EXPLANATORY VARIABLES: EXAMPLE Q = 1 + 2 L+ 3 K + u 1 + 2L + 3K + u u 1 + 3K pure K effect pure L effect 1 1 + 2L + 3K combined effect of L and K 1 + 2L Q K L The final element of the model is the disturbance term, u This causes the actual values of Q to deviate from the plane In this observation, u happens to have a positive value MULTIPLE REGRESSION WITH TWO EXPLANATORY VARIABLES: EXAMPLE Yi     X i   X i  ui Yˆi  b1  b2 X i  b3 X i ei  Yi  Yˆi  Yi  b1  b2 X i  b3 X i RSS   e i2   (Yi  b1  b2 X i  b3 X i ) The regression coefficients are derived using the same least squares principle used in simple regression analysis The fitted value of Y in observation i depends on our choice of b1, b2, and b3 The residual ei in observation i is the difference between the actual and fitted values of Y We define RSS, the sum of the squares of the residuals, and choose b1, b2, and b3 so as to minimize it, using first order condition MULTIPLE REGRESSION WITH TWO EXPLANATORY VARIABLES: EXAMPLE Model 3: OLS, using observations 1899-1922 (T = 24) Dependent variable: q coefficient std error t-ratio p-value -const -4,85518 14,5403 -0,3339 0,7418 l 0,916609 0,149560 6,129 4,42e-06 *** k 0,158596 0,0416823 3,805 0,0010 *** Mean dependent var 165,9167 Sum squared resid 2534,226 R-squared 0,942443 F(2, 21) 171,9278 Log-likelihood -89,96960 Schwarz criterion 189,4734 rho 0,098491 S.D dependent var 43,75318 S.E of regression 10,98533 Adjusted R-squared 0,936961 P-value(F) 9,57e-14 Akaike criterion 185,9392 Hannan-Quinn 186,8768 Durbin-Watson 1,535082 PROPERTIES OF THE MULTIPLE REGRESSION COEFFICIENTS A.1: The model is linear in parameters and correctly specified Y     X    k X k  u A.2: There does not exist an exact linear relationship among the regressors in the sample (No multicolinearity) A.3 The disturbance term has zero expectation A.4 The disturbance term is homoskedastic A.5 The values of the disturbance term have independent distributions A.6 The disturbance term has a normal distribution Provided that the regression model assumptions are valid, the OLS estimators in the multiple regression model are unbiased and efficient, as in the simple regression model MULTICOLINEARITY  Example: X1 - X2 =0  Definition: X1 and X2 are perfectly multi-collinear if there exists b1, b2 such that:  b1X1+b2X2 = a ( a: constant)  at least one of (bi) is non-zero  X1 and X2 are perfectly multi-collinear iff r(X1, X2) = +/-  X1 and X2 are highly multi-collinear if r(X1, X2) is large  ~X1 and X2 are highly multi-collinear if R2 of the model X1 on (a1, X2) is large 10 MULTICOLLINEARITY  Definition: X1, ,Xk are perfectly multi-colinear if there exists b1, ,bk :  b1X1+ +bkXk = a ( a: constant)  at least one of (bi) is non-zero  X1, ,Xk are highly multi-collinear R2 of the model (Xj on the rest and a1) is large  Assumption 6: no perfect multi-collinearity among X2, ,Xk  11 MULTICOLLINEARITY What would happen if you tried to run a regression when there is an exact linear relationship among the explanatory variables? The coefficient is not defined Q  1   L   K   K  u MULTICOLLINEARITY Model 4: OLS, using observations 1899-1922 (T = 24) Dependent variable: q coefficient std error t-ratio p-value const -10,7774 16,4164 -0,6565 0,5190 l 0,822744 0,190860 4,311 0,0003 *** k 0,312205 0,195927 1,593 0,1267 sq_k -0,000249224 0,000310481 -0,8027 0,4316 Mean dependent var 165,9167 Sum squared resid 2455,130 R-squared 0,944239 F(3, 20) 112,8920 Log-likelihood -89,58910 Schwarz criterion 191,8904 rho -0,083426 13 S.D dependent var 43,75318 S.E of regression 11,07955 Adjusted R-squared 0,935875 P-value(F) 1,05e-12 Akaike criterion 187,1782 Hannan-Quinn 188,4284 Durbin-Watson 1,737618 ... 189,4 734 rho 0,098491 S.D dependent var 43, 7 531 8 S.E of regression 10,98 533 Adjusted R-squared 0, 936 961 P-value(F) 9,57e-14 Akaike criterion 185, 939 2 Hannan-Quinn 186,8768 Durbin-Watson 1, 535 082... 0,190860 4 ,31 1 0,00 03 *** k 0 ,31 2205 0,195927 1,5 93 0,1267 sq_k -0,000249224 0,00 031 0481 -0,8027 0, 431 6 Mean dependent var 165,9167 Sum squared resid 2455, 130 R-squared 0,944 239 F (3, 20) 112,8920... -4,85518 14,54 03 -0 ,33 39 0,7418 l 0,916609 0,149560 6,129 4,42e-06 *** k 0,158596 0,04168 23 3,805 0,0010 *** Mean dependent var 165,9167 Sum squared resid 2 534 ,226 R-squared 0,9424 43 F(2, 21) 171,9278